The current version of Split Raster tool allows you to assign a base name prefix which it then increments from zero. This is not actually that helpful as you lose the connection with the polygon that clipped it. For example in the UK our national grid splits up into 100Km cells which have a two letter code. This is important to us Brits! If I split up a raster covering the UK I end up with for example r0, r1, r2 ,r3. I would rather see rSY, rSX,rTV,rTQ....
The tool allows you to choose a FeatureClass to split by and a couple of other related options.
My idea is to add a new parameter that allows you to select a field in your splitting polygon layer which would be used to build the output raster dataset name. In my example it would be a text field holding the two letter grid cell code.
A work around to this is to abandon Split Raster and use an iterator feeding into a clip tool. The BIG disadvantage of this is that you lose the tools ability to parallelize the problem and take advantage of model multi-core machines.
Adding such a parameter to the Split Raster tool is such an obvious addition especially for Pro and ideally ArcMap.